How to Find the Circumradius of a Triangle Using Formula and Sides
A triangle has three sides, two of those sides are called the base and the height. The perimeter of a triangle is the sum of the lengths of all the sides. If a triangle has three equal sides, then the perimeter equals three times the side length.
Circumradius of a Circle
The circumradius is the radius of a circle that will pass through all three vertices of a triangle. A triangle is surrounded by a single circle that is perpendicular to the base of the triangle. This is true for an equilateral triangle and any other type of triangle.
What is Circumcircle?
A circumcircle is a circle that circumscribes any polygon and touches each polygon vertex. The radius of the circumcircle is the distance between the point of meeting of bisectors and the vertex. The circumcircle is drawn by taking the meeting point of the perpendicular bisector to the sides of the polygon as the centre.
What is Circumradius?
The circumradius of a polygon or a triangle is the radius of the circle that circumscribes (surrounds) that polygon or triangle. It is the distance from the centre of the circumcircle to one of the vertices of the polygon or triangle. In the case of a triangle, the circumradius is also known as the circumcenter radius.
How to Find Circumradius and What is the Formula?
Let that there is a circumcircle circumscribing a triangle with sides a, b and c with a circumcentre.
$2 \mathrm{R}=\dfrac{a}{\operatorname{Sin} A}=\dfrac{b}{\operatorname{Sin} B}=\dfrac{c}{\sin C}$
Now, Area of triangle $\left(A_t\right)=\dfrac{1}{2} a b \operatorname{Sin} C$
$\operatorname{Sin} C=\dfrac{c}{2 R}$
Therefore, $\mathrm{A}_{\mathrm{t}}=\dfrac{1}{2} a b \dfrac{c}{2 R}$.
Now, $\mathrm{R}=\dfrac{a b c}{4 A}$
Hence, the required formula of circumradius of the triangle is $\mathrm{R}=\dfrac{a b c}{4 A}$.
Solved Examples
1. What is the circumradius of a triangle with side lengths of 4, 5, and 6 units?
The circumradius of a triangle can be found using the formula:
$\mathrm{R}=\dfrac{a b c}{4 A}$.
where R is the circumradius, a, b, and c are the side lengths of the triangle, and A is the area of the triangle.
In this case, we have a triangle with side lengths of 4, 5, and 6 units.
To find the area of the triangle, we can use Heron's Formula:
$\text { Area } =\sqrt{s(s-a)(s-b)(s-c)}$
$\text { Area } =\text { area }$
$S =\text { semi-perimeter }$
$a=\text { length of side a }$
$b=\text { length of side } b$
$c=\text { length of side } c$
Where S is the semiperimeter of the triangle.
S = $\dfrac{(4+5+6)}{2} = 7.5$
$A=\sqrt{(7.5(7.5-4)(7.5-5)(7.5-6))}$
$A=\sqrt{(7.5(3.5)(2.5)(1.5))}$
$A=\sqrt{98.43}$
$=9.92$
So, we can plug the values into the formula:
$\mathrm{R}=\dfrac{a b c}{4 A}$
$= \dfrac{4\times 5\times 6}{4 \times 9.92} = 3.02$
Therefore, the circumradius of the triangle with side lengths of 4, 5, and 6 units is 3.02 units.
2. Triangle ΔABC is inscribed in a circle O, and side AC passes through the circle’s centre. Find the circle’s diameter.
We know that the triangle inscribed by a chord that passes through the centre of the circle is a right triangle.
Given, BC = 16 and AB = 12.
Hypotenuse theorem can be applied here,
$A C^{2}=A B^{2}+B C^{2}$
$=A C^{2}=12^{2}+16^{2}$
$=A C=\sqrt{12^{2}+16^{2}}$
$=A C=\sqrt{144+256}$
$= AC=\sqrt{400}$
$=A C=20$
The diameter of the circle is 20.
3. What is the area of a circle that is inscribed in a square of area 64 square units?
Inscribed circles in squares have a diameter equal to their sides when the circle is encircled. As a result, the radius of the inscribed circle is 4 units, while the diameter is 8 units.
The area of a circle of radius r units is $\mathrm{~A}=\pi \mathrm{r}^2$.
Substitute r=4 in the formula.
$\mathrm{~A}=\pi \mathrm{4}^2 = 16 \pi$≈50.24
Therefore, the area of the inscribed circle is about 50.24 square units.
Practice Questions
1. The circumradius of the triangle, whose sides are 13,12 and 5, is
A) 15
B) 132
C) 6
Ans: 6 (Option C)
2. The in-radius and circumradius of a right-angled triangle are 3 cm and 12.5 cm, respectively. The area of the triangle is:
A) 84
B) 64
C) 94
Ans: 84 (Option A)
3. The in-radius and circumradius of a right-angled triangle are 2 cm and 10 cm, respectively. The area of the triangle is:
A) 4
B) 8
C) 13.5
Ans: 13.5 (Option C)
Summary
The basic definition of circumradius in geometry is that it refers to the radius of a circle that passes through all three vertices of a triangle. In simple words, it is the radius of a circle that circumscribes the triangle. A triangle also has one circumscribed circle around it, which is perpendicular to its base. This is true for an equilateral triangle and any other type of triangle.
In conclusion, you can use the Law of Sines and the Law of Cosines to find the circumradius of a triangle. Do not forget to use the formula, $\mathrm{R}=\dfrac{a}{\operatorname{Sin} A}$ and you will be able to determine the value of A.
FAQs on Circumradius of a Triangle Explained with Formula
1. What is the circumradius of a triangle?
The circumradius of a triangle is the radius of the circle that passes through all three vertices of the triangle, called the circumcircle. It is usually denoted by R. The center of this circle is the circumcenter, which is the point where the perpendicular bisectors of the sides intersect. The circumradius depends on the side lengths and angles of the triangle.
2. What is the formula for the circumradius of a triangle?
The formula for the circumradius of a triangle is R = abc / 4Δ, where a, b, and c are the side lengths and Δ is the area of the triangle. Another common formula is R = a / (2 sin A), where a is a side and A is the angle opposite it. These formulas are widely used in trigonometry and coordinate geometry.
3. How do you find the circumradius using the sides of a triangle?
You can find the circumradius using the formula R = abc / 4Δ after calculating the area of the triangle. Follow these steps:
- Step 1: Find the semi-perimeter: s = (a + b + c)/2.
- Step 2: Use Heron’s formula to find area: Δ = √[s(s-a)(s-b)(s-c)].
- Step 3: Substitute into R = abc / 4Δ.
4. What is the circumradius of an equilateral triangle?
The circumradius of an equilateral triangle with side length a is R = a / √3. Since all sides and angles (60°) are equal, the circumcenter lies at the center of the triangle. For example, if a = 6, then R = 6 / √3 = 2√3.
5. What is the circumradius of a right triangle?
The circumradius of a right triangle is equal to half of the hypotenuse. If the hypotenuse is c, then R = c/2. This happens because the circumcenter lies at the midpoint of the hypotenuse. For example, if the hypotenuse is 10, then R = 5.
6. How do you find the circumradius using trigonometry?
You can find the circumradius using the trigonometric formula R = a / (2 sin A). Here:
- a is a side of the triangle
- A is the angle opposite that side
7. Where is the circumcenter located in different types of triangles?
The circumcenter location depends on the type of triangle.
- In an acute triangle, it lies inside the triangle.
- In a right triangle, it lies at the midpoint of the hypotenuse.
- In an obtuse triangle, it lies outside the triangle.
8. What is the relationship between circumradius and area of a triangle?
The relationship between circumradius and area is given by Δ = abc / 4R, where Δ is the area and a, b, c are the side lengths. Rearranging gives R = abc / 4Δ. This formula connects geometry (area) with circle properties (circumcircle).
9. Can you give a worked example of finding the circumradius?
Yes, here is a worked example using sides 3, 4, and 5.
- This is a right triangle with hypotenuse 5.
- For a right triangle, R = hypotenuse/2.
- So, R = 5/2 = 2.5.
10. What is the difference between circumradius and inradius of a triangle?
The circumradius is the radius of the circle passing through all vertices, while the inradius is the radius of the circle inscribed inside the triangle touching all sides. Key differences:
- Circumradius formula: R = abc / 4Δ
- Inradius formula: r = Δ / s (where s is semi-perimeter)
- Circumcenter uses perpendicular bisectors; incenter uses angle bisectors.
